Algebraic And Analytic Integrability Of Generalized Lorenz System

It is well known that integrability has been an old but significant topic in the fieldof differential equations. In general, there are many ways to discuss the integrabil-ity of ordinary differential equations. This paper provide primarily concerned withthe algebra integrability and local analytic integrability of polynomial differential sys-tems(especially generalized Lorenz systems). On algebraic integrability, we mainlyconcern Darboux integrability, which is determined by Darboux polynomials. But tillnow there is no a definite procedure to built the first integral for general differentialequations, even we consider the invariant polynomial first integrals for polynomial d-ifferential equations. In this paper we will study a class of generalized Lorenz system,and discuss its Darboux polynomials and local analytic first integrals at the origin.Calculating polynomial systems of Darboux polynomials is a classical and diffi-cult problem. Studying about Darboux polynomials of Lorenz system, Segur[20] andKu′s[15] got three independent invariant polynomials respectively. Later, Llibre andZhang[10] proved that these invariant polynomials is complete. In2002, Peter intro-duced the weight of polynomials to simplify the calculation of Darboux polynomials.In this paper, we will use the method of [9] to calculate the Darboux polynomials ofthe generalized Lorenz system. We obtain all parameter conditions for which the gen-eralized Lorenz systems have a Darboux polynomials and provide the correspondingDarboux polynomials.Calculating the first integrals of polynomial systems is very difficult. Poincare′[17]was the first one to study this problem. He proved that if the eigenvalues of the lin-ear part of analytic differential systems at the singularities do not satisfy any resonantconditions, then the system has no analytic first integrals in the neighborhood of thesingularities. Li, Llibre and Zhang [8] extended and improved the Poincare′'s result. In 2005, Llibre and Valls gave the two necessary conditions on existence of analytic firstintegral at the origin. In this paper, we study analytic first integral of a class Lorenzsystem at the origin. by using the classification of eigenvalues of linearized system atthe origin.In addition, for the three-dimensional quadratic polynomial differential system-s, we provide a necessary condition for the existence of form first integral when thecorresponding linearize system has three non-zero real eigenvalues, and a unique ho-mogeneous polynomial first integral.